A Hermitian Calabi-Yau manifold does not always possess a Kähler metric. There are many examples of such manifolds: Calabi-Yau manifolds obtained from the conifold transition (see [1]}), hyperkähler fibrations over Riemannian surfaces ([2]}, [3]}), factors of Lie groups, etc. A less stringent notion than Kähler is that of balanced metric, introduced by Michelson
[4]}. These are metrics satisfying \(d\omega ^{n-1}=0\) . Closely related is the notion of conformally balanced metric, defined by the condition
\(d(\Vert \Omega \Vert _\omega \omega ^{n-1}) = 0\) . The conformally balanced condition was recognized by Li and Yau [5]} as one of the equations in the Hull-Strominger system in string theory. In terms of the metric \(\eta = \Vert \Omega \Vert \omega \) , it can also be expressed as \(d(\Vert \Omega \Vert ^2 \eta ^{n-1})= 0\) .
